arXiv:math-ph/0302009v1 4 Feb 2003
Equivalence Principle and the Principle of Local Lorentz Invariance∗ W. A. Rodrigues, Jr.(1,2) †and M. Sharif(1)‡ (1)
Institute of Mathematics, Statistics and Scientific Computation IMECC–UNICAMP CP 6065 13083-970 Campinas, SP, Brazil and (2)
Wernher von Braun
Advanced Research Center, UNISAL Av. A. Garret, 257 13087-290 Campinas, SP Brazil
pacs: 04.90+e 03.30+p 02/04/2003
Contents 1 Introduction
2
2 Some Basic Definitions 2.1 Relativistic Spacetime Theories . . . . . . . . . . . . . . . . . . 2.2 Reference Frames . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Pseudo Inertial Reference Frames . . . . . . . . . . . . . 2.2.2 Naturally Adapted Charts to a Given Reference Frame . 2.2.3 Local Lorentzian Coordinate Chart . . . . . . . . . . . . 2.2.4 Local Lorentz Reference Frame . . . . . . . . . . . . . .
. . . . . .
3 3 4 5 6 6 7
3 Physically Equivalent Reference Frames
8
4 LLRF γs and the Equivalence Principle
11
∗ published: Found. of Physics 31, 1785-1806 (2001). Includes corrigenda published Found. Phys. 32, 811-812 (2002). † e-mail:
[email protected] ‡ Permanent Address: Department of Mathematics, Punjab University, Quaid-e-Azam Campus Lahore-54590, PAKISTAN, e-mail:
[email protected]
1
5 Physical non equivalence of PIRFs V and Z on a Friedmann Universe 13 5.1 Mechanical experiments distinguish PIRFs . . . . . . . . . . . . 15 6 LLRF γ and LLRF γ ′ are not Physically Equivalent on a Friedmann Universe. 17 7 Conclusions
18 Abstract
In this paper we scrutinize the so called Principle of Local Lorentz Invariance (PLLI ) that many authors claim to follow from the Equivalence Principle. Using rigourous mathematics we introduce in the General Theory of Relativity two classes of reference frames (PIRFs and LLRF γs) which natural generalizations of the concept of the inertial reference frames of the Special Relativity Theroy. We show that it is the class of the LLRF γs that is associated with the PLLI. Next we give a defintion of physically equivalent referefrence frames. Then, we prove that there are models of General Relativity Theory (in particular on a Friedmann universe) where the PLLI is false. However our find is not in contradiction with the many experimental claims vindicating the PLLI, because theses experiments do not have enough accuracy to detect the effect we found. We prove moreover that PIRFs are not physically equivalent.
1
Introduction
In this paper we scrutinize the so called Principle of Local Lorentz Invariance (PLLI ) that some authors claim to follow from the Equivalence Principle (EP ). We show that PLLI is false according to General Theory of Relativity (GRT ), but nevertheless it is a very good approximation in the physical world we live in. In order to prove our claim, we recall the mathematical definition of reference frames in GRT which are modelled as certain unit timelike vector fields. We study the classification of reference frames and give a physically motivated and mathematical rigorous definition of physically equivalent reference frames. We investigate next which are the reference frames in GRT which share some of the properties of the inertial reference frames (IRFs) of the Special Theory of Relativity (SRT). We found that there are two kind of frames that appear as generalizations of the IRFs of SRT. These are the pseudo inertial reference frames (PIRFs) and the local Lorentz reference frames (LLRF γs). Now, PLLI is a statement that LLRF γs are physically equivalent. We show that PLLI is false by expliciting showing that there are models of GRT (explicitly a Friedmann Universe) containing LLRF γs which are not physically equivalent. We emphasize that our finding is not in contradiction with the many experimental proofs offered as vindicating the PLLI, since all this proofs do not have enough accuracy to detect the effect we have found which is proportional to 2av 2 , where a is a future pointing time-like curve γ : R ⊃ I → M such that g(γ∗ u, γ∗ u) = 1. The inclusion parameter I → R in this case is called the proper time along γ, which is said to be the world line of the observer. Observation 1. The physical meaning of proper time is discussed in details, e.g., in [5,6] which deals with the theory of time in relativistic theories. Definition 4. An instantaneous observer is an element of T M , i.e., a pair (z, Z), where z ∈ M , and Z ∈ Tz M is a future pointing unit time-like vector. The Proposition 1 together with the above definitions suggests: Definition 5. A reference frame in ST =< M, D, g > is a time-like vector field which is a section of T U, U ⊆ M such that each one of its integral lines is an observer. Observation 2. In [4-6] an arbitrary reference frame Q ∈ secT U ⊆ sec T M is classified as (i), (ii) below. (i) according to its synchronizability. Let αQ = g(Q, ). We say that Q is locally synchronizable iff αQ ∧ dαQ = 0. Q is said to be locally proper time synchronizable iff dαQ = 0. Q is said to be synchronizable iff there are C ∞ functions h, t : M → R such that αQ = hdt and h > 0. Q is proper time synchronizable iff αQ = dt. These definitions are very intuitive. (ii) according to the decomposition of 1 DαQ = aQ ⊗ αQ + ωQ + σQ + ΘQ h, 3 4
(2)
where h = g − αQ ⊗ αQ
(3)
is called the projection tensor (and gives the metric of the rest space of an instantaneous observer [17-19]), aQ is the (form) acceleration of Q, ωQ is the rotation of Q, σQ is the shear of Q and ΘQ is the expansion ratio of Q . In a coordinate chart (U, xµ ), writing Q = Qµ ∂/∂xµ and h = (gµν −Qµ Qν )dxµ ⊗dxν we have ωQµν = Q[µ;ν] , 1 σQαβ = [Q(µ;ν) − ΘQ hµν ]hµα hνβ , 3 ΘQ = Qµ ;µ .
(4)
We shall need in what follows the following result that can be easily proved: αQ ∧ dαQ = 0 ⇔ ωQ = 0.
(5)
Eq.(3) means that rotating reference frames (i.e., frames for which ωQ 6= 0) are not locally synchronizable. This result is the key in order to solve the misconceptions usually associated with rotating reference frames even in the SRT (see [8] for examples). Observation 3. In Special Relativity where the space time manifold is < M = R4 , g = η, Dη >2 an inertial reference frame (IRF ) I ∈ sec T M is defined by Dη I = 0. We can show very easily that in GRT where each gravitational field is modelled by a spacetime < M, g, D > there are no frame Q ∈ sec T M satisfying DQ = 0. So, no IRF exist in any model of GRT. The following question arises naturally: which characteristics a reference frame on a GRT spacetime model must have in order to reflect as much as possible the properties of an IRF of SRT ? The answer to the question is that there are two kind of frames in GRT [PIRFs (definition 6) and LLRFs (definition 9)], such that each frame in one of these classes share some important aspects of the IRFs of SRT. Both concepts are important and as we will see, it is important to distinguish between them in order to avoid misunderstandings. 2.2.1
Pseudo Inertial Reference Frames
Definition 6. A reference frame I ∈ sec T U, U ⊂ M is said to be a pseudo inertial reference frame (PIRF ) if DI I = 0 and αI ∧ dαI = 0, and αI = g(I, ). This definition means that a PIRF is in free fall and is non rotating. It means also that it is at least locally synchronizable. 2 η is a constant metric, i.e., there exists a chart hxµ i of M = R4 such that η(∂/∂xµ , ∂/∂xν ) = ηµν , the numbers ηµν forming a diagonal matrix with entries (1, −1, −1, −1). Also, D η is the Levi-Civita connection of η.
5
2.2.2
Naturally Adapted Charts to a Given Reference Frame
Definition 7. Let Q ∈ sec T U, U ⊆ M be a reference frame. A chart in U of the maximal oriented atlas of M with coordinate functions hxµ i such that ∂/∂x0 ∈ sec T U is a timelike vector field and the ∂/∂xi ∈ sec T U (i = 1, 2, 3) are spacelike vector fields is said to be a possible naturally adapted coordinate chart to the frame Q (denoted (nacs—Q) in what follows) if the space-like components of Q are null in the natural coordinate basis h∂/∂xµ i of T U associated with the chart.3 2.2.3
Local Lorentzian Coordinate Chart
Definition 8. A chart (U, ξ µ ) of the maximal oriented atlas of M is said to be a local Lorentzian coordinate chart (LLCC) and hξ µ i are said to be local Lorentz coordinates (LLC ) in p0 ∈ U iff g(∂/∂ξ µ , ∂/∂ξ ν ) |p0 = ηµν ,
µ Γα βµ (ξ ) |p0 = 0,
1 α µ µ α µ Γα βγ,µ (ξ ) |p = − (Rβγµ (ξ ) + Rγβµ (ξ )) |p , p 6= p0 3
(6)
(7)
Let (V, xµ ) (V ∩ U 6= ∅) be an arbitrary chart. Then, supposing that p0 is at the origin of both coordinate systems the following relations holds (approximately) 1 ξ µ = xµ + Γµαβ (p0 )xα xβ , 2 1 xµ = ξ µ − Γµαβ (p0 )ξ α ξ β , 2
(8)
where in eqs.(8) Γµαβ (p0 ) are the values of the connection coefficients at p0 expressed in the chart (V, xµ ). The coordinates hξ µ i are also known as Riemann normal coordinates and the explicit methods for obtaining them are described in many texts of Riemaniann geometry as e.g., [9,10] and of GRT, as e.g., [11,12]. Observation 4. Let γ ∈ U ⊂ M be the world line of an observer in geodetic motion in spacetime, i.e., Dγ∗ γ∗ = 0. Then as it is well known [11] we can introduce in U a LLC hξ µ i such that for every p ∈ γ we have ∂ = γ∗|p ; g(∂/∂ξ µ , ∂/∂ξ ν )|p∈γ = ηµν , ∂ξ 0 p∈γ Γµνρ (ξ µ ) p∈γ = g µα g(∂/∂ξ α , D∂/∂ξν ∂/∂ξ ρ ) p∈γ = 0.
(9)
Take into account for future reference that if the < ξ µ > are LLC then it is clear from definition 8 that in general Γνµρ (ξ µ ) |p 6= 0 for all p ∈ / γ. 3 We
can be prove very easily that there is an infinity of different (nacs—Q).
6
2.2.4
Local Lorentz Reference Frame
Definition 9. Given a geodetic line γ ⊂ U ⊂ M and LLCC (U, ξ µ ) we say that reference frame L =∂/∂ξ 0 ∈ sec T U is a Local Lorentz reference frame associated to γ (LLRF γ)4 iff ∂ L|p∈γ = = γ∗ |p , ∂ξ 0 p∈γ αL ∧ dαL |p∈γ = 0.
(10)
Moreover, we say also that the Riemann normal coordinate functions or Lorentz coordinate functions (LLC ) < ξ µ > are associated with the LLRF γ. Observation 5. It is very important to have in mind that for a LLRF γ L in general DL L|p∈γ / 6= 0 (i.e., only the integral line γ of L in free fall in general), and also eventually αL ∧ dαL |p∈γ 6= 0, which may be a surprising / result for many readers. In contrast, a PIRF I such that I|γ = L|γ has all its integral lines in free fall and the rotation of the frame is always null in all points where the frame is defined. Finally its is worth to recall that both I and L may eventually have shear and expansion even at the points of the geodesic line γ that they have in common. This last point will be important in our analysis of the PLLI in section 6. Definition 10. Let γ be a geodetic line as in definition 9. A section s of the orthogonal frame bundle F U, U ⊂ M is called an inertial moving frame along γ (IMFγ) when the set sγ = {(e0 (p), e1 (p), e2 (p), e3 (p)), p ∈ γ ∩ U } ⊂ s,
(11)
it such that ∀p ∈ γ e0 (p) = γ∗ |p , g(eµ , eν )|p∈γ = ηµν
(12)
Γµνρ (p) = g µα g(eα (p), Deν (p) eρ (p)) = 0
(13)
with Observation 6. The existence of s ∈ sec F U satisfying the above conditions can be easily proved [9]. Introduce coordinate functions < ξ µ > for U such that at p0 ∈ γ, e0 (p0 ) = ∂ξ∂ 0 = γ∗|p0 , and ei (p0 ) = ∂ξ∂ i , i = 1, 2, 3 (three po
po
orthonormal vectors) satisfying Eq.(9 ) and parallel transport the set eµ (p0 ) along γ. The set eµ (p0 ) will then also be Fermi transported [4] since γ is a geodesic and as such they define the standard of no rotation along γ. Observation 7. Let I ∈ sec T V be a PIRF and γ ⊂ U ⊂ V one of its integral lines and let < ξ µ >, U ⊂ M be a LLC through all the points of the world line γ such that γ∗ = I|γ . Then, in general < ξ µ > is not a (nacs|I) in 0 U , i.e., I|p∈γ even if I|p∈γ = ∂/∂ξ 0 p∈γ . / 6= ∂/∂ξ p∈γ / Observation 8. Before concluding this section it is very much important to recall again that a reference frame field as introduced above is a mathematical 4 When
no confusion arises and γ is clear from the context we simply write LLRF.
7
instrument. It did not necessarily need to have a material substratum (i.e., to be realized as a material physical system) in the points of the spacetime manifold where it is defined. More properly, we state that the integral lines of the vector field representing a given reference frame do not need to correspond to worldlines of real particles. If this crucial aspect is not taken into account we may incur in serious misunderstandings. We observe moreover that the concept of reference frame fields has been also used since a long time ago by Matolsci [13], although this author uses a somewhat different terminology.
3
Physically Equivalent Reference Frames
The objective of this section is two recall the definition of physically equivalent reference frames in a spacetime theory and in particular in GRT [1] which will be used in section 6 to prove that the PLLI is false. In order to do that we need to recall some definitions. Let hM, D, gi be a Lorentzian spacetime and let GM be the group of all diffeomorfisms of M , called the manifold mapping group. Let A ⊆ M . Definition 11. The diffeomorfism GM ∋ h : A → M induces a deforming mapping ¯ h∗ : T 7→ h∗ T = T (14) such that, (i) If f : M ⊇ A → R, then h∗ f = f ◦ h−1 : h(A) → R.
(15)
(ii) If T sec T (r,s) (A) ⊆ sec T (M ), where T (r,s) (A) is the sub-bundle of tensors of the type (r, s) of the tensor bundle T (M ),then (h∗ T)he (h∗ ω1 , ..., h∗ ωr , h∗ X1 , ..., h∗ Xs ) Te (ω1 , ..., ωr , X1 , ..., Xs )
(16)
∀Xi ∈ sec Te (A), i = 1, 2, ..., r, ∀ωj ∈ sec T ∗ A, j = 1, 2, ..., s, ∀e ∈ M . (iii) If D is the Levi-Civita connection of g on M and X, Y ∈ sec T M , then (h∗ Dh∗ X h∗ Y )he h∗ f = (DX Y )e f, ∀e ∈ M h∗ Dh∗ X h∗ Y ≡ h∗ (DX Y ).
(17)
If {fµ = ∂/∂xµ } is a coordinate basis for T A and {θµ = dxµ } is the corresponding dual basis for T ∗ A and if ...µr ν1 θ ⊗ ... ⊗ θνs ⊗ fµ1 ⊗ ... ⊗ fµr , T = Tνµ11....ν s
(18)
then ...µr ◦ h−1 )h∗ θν1 ⊗ ... ⊗ h∗ θνs ⊗ h∗ fµ1 ⊗ ... ⊗ h∗ fµr . h∗ T = (Tνµ11....ν s
8
(19)
Suppose now that A and h(A) can be covered by the local chart (U, ϕ) of the maximal atlas of M , and that A ⊆ U, h(A) ⊆ U . Let hxµ i be coordinate functions associated with (U, ϕ). The mapping x′µ = xµ ◦ h−1 : h(U ) → R
(20)
defines a coordinate transformation hxµ i 7→ hx′µ i if h(U ) ⊇ A ∪ h(A). Indeed, hx′µ i are the coordinate functions associated with a local chart (V, χ) where h(U ) ⊆ V and U ∩V 6= ∅. Now, since under these conditions h∗ ∂/∂xµ = ∂/∂x′µ ′ and h∗ dxµ = dx µ , eqs.(19) and (20) imply that (h∗ T)hx′µ i (he) = Thxµ i (e).
(21)
...µr (xµ (e)) are the components of T in the local coorIn eq.(21) Thxµ i (e) ≡ Tνµ11....ν s µ µ 1 ...µr dinate basis {∂/∂x }, {dx } at event e ∈ M , and (h∗ T)hx′µ i (he) ≡ T¯ν′µ1 ....ν (x′µ (he)) s ¯ are the components of T = h∗ T in the local coordinate basis {h∗ ∂/∂xµ = ′ ∂/∂x}, {h∗dxµ = dx µ } at the point he. Then eq.(21) reads ...µr 1 ...µr (xµ (e)). (x′µ (he)) = Tνµ11....ν T¯ν′µ1 ....ν s s
(22)
Using eq.(20) we can also write 1 ...αr 1 ...µr (x′µ (h−1 e)) (x′µ (e)) = (Λ−1 )µα11 ...(Λ)βνss Tβ′α1 ....β T¯ν′µ1 ....ν s s
(23)
where Λµα = ∂x′µ /∂xα , etc. Definition 12. Let h ∈ GM . If for a geometrical object T we have h∗ T = T
(24)
then h is said to be a symmetry of T and the set of all {h ∈ GM } such that eq.(24) holds is said to be the symmetry group of T. Definition 13. Let Υ, Υ′ ∈ Mod τ , Υ = (M, D, g, T1 , . . . , Tm ) and Υ′ = (M, D′ , g ′ , T′1 , . . . , T′m ) with the Ti , i = 1, . . . , m defined in U ⊆ M and T′i , i = 1, . . . , m defined in V ⊆ M . We say that Υ is equivalent toΥ′ (and denotes Υ ∼ Υ′ ) if there exists h ∈ GM such that Υ′ = h∗ Υ, i.e., V ⊆ h(U ) and D′ = h∗ D, g ′ = h∗ g, T′1 = h∗ T1 , ..., T′m = h∗ Tm
(25)
Theories satisfying definition 14 are called generally covariant and Υ, Υ′ ∈ Mod τ represent indeed the same physical model. ¯ ∈ Mod τ, Υ = (M, D, g, T1 , . . . , Tm ), Υ ¯ = Definition 14. Let Υ, Υ (M, h∗ D, h∗ g, h∗ T1 , . . . , h∗ Tm ) with the Ti , i = 1, . . . , m defined in U ⊆ M and T′i , i = 1, . . . , m defined in V ⊆ h(U ) ⊆ M and such that D = h∗ D, g = h∗ g.
(26)
¯ is said to be the h-deformed version of Υ. Then Υ ¯ ∈ sec T V ⊆ sec T M , U ∩ V 6= Definition 15. Let Q ∈ sec T U ⊆ sec T M, Q µ µ ∅ and let hx i , h¯ x i (the coordinate functions associated respectively to the 9
¯ and suppose charts (U, ϕ) and (V, ϕ)) ¯ be respectively a (nacs|Q) and a (nacs|Q) ¯ ∗ Q and Q ¯ ¯ = h ¯ is said to be a hthat x¯µ = xµ ◦ ¯ h−1 : ¯ h(U ) → R. Thus, Q deformed version of Q. ¯ ∈ Mod τ be as in definition 14. Call o = (D, g, T1 , . . . , Tm ) and Let Υ, Υ o¯ = (D, g, h∗ T1 , . . . , h∗ Tm ). Now, o is such that it solves a set of differential µ equations in ϕ(U ) ⊂ R4 with a given set of boundary conditions denoted bohx i , which we write as ohx α Dhx µ i (ohxµ i )e = 0 ; b
µ
i
; e ∈ U,
(27)
¯ ) ⊆ V solves and o¯ defined in h(U α ¯ xµ i )|he = 0 ; bh¯ ∗ oh¯xµ i ; h ¯ e ∈ h(U ¯ ) ⊆ V. Dh¯ xµ i (h∗ oh¯
(28)
α α mean α = 1, 2, . . . , m sets of differential In eqs.(27) and (28) Dhx µ i and D hx′ µ i
equations in R4 . ¯ ∈ Mod τ are deformed How can an observers living on M discover that Υ, Υ versions of each other? In order to answer this question we need additional definitions. ¯ be as in definition 15. We say that Q and Q ¯ are Definition 16. Let Q, Q ¯ physically equivalent according to theory τ (and we denote Q ∼ Q) iff (i)
¯ DQ = DQ
(29)
and (ii) the system of differential equations (27) must have the same functional µ ¯ form as the system of differential equations (28) and bh∗ oh¯x i must be relative µ ohxµ i µ ohxµ i to h¯ x i the same as b is relative to hx i and if b is physically realizable µ ¯ then bh∗ oh¯x i must also be physically realizable. Definition 17. Given a reference frame Q ∈ sec T U ⊆ sec T M the set of all diffeomorfisms {h ∈ GM } such that h∗ Q ∼ Q forms a subgroup of GM called the equivalence group of the class of reference frames of kind Q according to the theory τ . Observation 9. We can easily verify using definitions 16 and 17 any two IRF in Minkowski space time (M, Dη , η) (observation 3) are equivalent and that the equivalence group of the class of inertial reference frames is the Poincar´e group. Of course, we can verify that the symmetry group (definition 12) of Dη and η is also the Poincar´e group. It is the existence of this symmetry group that permits a mathematical definition of the Special Principle of Relativity.5 We can also show without difficulties that two distinct rotating references frames (with have the same angular velocity relative to a given IRF and that have the same radius) are physically equivalent. Of course, no IRF is equivalent to any ¯ ∈ Mod rotating frame. A comprehensive example of phenomena related as Υ, Υ τ in definitions 14 and 15 is (in Minkowski spacetime) the electromagnetic field 5 See [14] where we point out that the definition of physically equivalent reference frames given above leads to contradictions in SRT if superluminal phenomena exist and we insist in mantaining the validity of the Special Principle of Relativity.
10
of a charge at rest relative to an IRF I and the field of a second charge in uniform motion relative to the same IRF I and its field relative to an IRF I′ where the second charge is at rest.
4
LLRF γs and the Equivalence Principle
There are many presentations of the EP and even very strong criticisms against it, the most famous being probably the one offered by Synge [15]. We are not going to bet on this particular issue. Our intention here is to prove that there are models of GRT where the so called Principle of Local Lorentz Invariance (PLLI) which according to several authors (see below) follows from the Equivalence Principle is not valid in general. Our strategy to prove this strong statement is to give a precise mathematical wording to the PLLI (which formalizes the PLLI as introduced by several authors) in terms of a physical equivalence of LLRF γs (see below) and then prove that PLLI is a false statement according to GRT. We start by recalling formulations and comments concerning the EP and the PLLI. According to Friedmann [16] the “Standard formulation of the EP characteristically obscure [the] crucial distinction between first order laws and second order laws by blurring the distinction between infinitesimal laws, holding at a single point, and local laws, holding on a neighborhood of a point”.... According to our point of view, in order to give a mathematically precise formulation of Einstein’s EP besides the distinctions mentioned above between infinitesimal and local laws, it is also necessary to distinguish between some very different (but related) concepts, namely, 6 (i) The concept of an observer (definition 1); (ii) The general concept of a reference frame in GRT (Definition 4); (iii) The concept of a natural adapted coordinate system to a reference frame (Definition 7); (iv) The concept of PIRFs (definition 6) and LLRF γs (definition 9) on U ⊂M ; (v) The concept of an inertial moving observer carrying a tetrad along γ (a geodetic curve), a concept we abbreviate by calling it an IMF γ (definition 10). Einstein’s EP is formulated by Misner, Thorne and Wheeler (MTW ) [17] as follows: “in any and every Local Lorentz Frame (LLF), anywhere and anytime in the universe, all the (non-gravitational) laws of physics must take on their familiar special relativistic forms. Equivalently, there is no way, by experiments confined to small regions of spacetime to distinguish one LLF in one region of spacetime from any other LLF in the same or any other region”. We comment here that these authors7 did not give a formal definition of a LLF. They try to make intelligible the EP by formulating its wording in terms of a LLCC 6 These concepts are in general used without distinction by different authors leading to misunderstandings and misconceptions. 7 For the best of our knowledge no author gave until now the fomal definition of a LLRF as in definition 9.
11
(see definition 8) and indeed these authors as many others do not distinguish the concept of a reference frame Z ∈ sec T M from that of a (nacs|Z). This may generate misunderstandings. The mathematical formalization of a LLF used by MTW (and many other authors) corresponds to the concept of LLRF introduced in definition 9. In [18] Ciufolini and Wheeler call the above statement of MTW the medium strong form of the EP. They introduced also what they called the strong EP as follows: “in a sufficiently small neighborhood of any spacetime event, in a locally falling frame, no gravitational effects are observable”. Again, no mathematical formalization of a locally falling frame is given, the formulation uses only LLCC 8 . Following [17,18] recently several authors as, e.g., Will [19], Bertotti and Grishchuk [20] and Gabriel and Haugan [21] (see also Weinberg [22] claim that Einstein EP requires a sort of local Lorentz invariance. This concept is stated in, e.g., [20] with the following arguments. To start we are told that to state the Einstein EP we need to consider a laboratory that falls freely through an external gravitational field, such that the laboratory is shielded, from external non-gravitational fields and is small enough such that effects due to the inhomogeneity of the field are negligible through its volume. Then, they say, that the local non-gravitational test experiments are experiments performed within such a laboratory and in which self-gravitational interactions play no significant part. They define: “The Einstein EP states that the outcomes of such experiments are independent of the velocity of the apparatus with which they are performed and when in the universe they are performed”. This statement is then called the Principle of Local Lorentz Invariance (PLLI ) and ‘convincing’ proofs of its validity are offered, and not need to be repeated here. Prugovecki [23] (pg 62) endorses the PLLI and also said that it can be experimentally verified. In his formulation he translates the statements of [16-22] in terms of Lorentz and Poincar´e covariance of measurements done in two different IMF γ (see Definition 10). Based on these past tentatives of formalization9 we give the following one. Einstein EP : Let γ be a timelike geodetic line on the world manifold M. For any LLRF γ (see definition 9) all nongravitational laws of physics, expressed through the coordinate functions hξ µ i which are LLC associated with the LLRF γ (definition 9) should at each point along γ be equal (up to terms in first order in those coordinates) to their special relativistic counterparts when 8 Again, no mathematical formalization of a locally falling frame is given, the formulation uses only the concept of LLCCs. Worse, if local means in a neighborhood of a given spacetime event this principle must be false. For, e.g., it is well known that the Riemann tensor couples locally with spinning particles. Moreover, the neigbourhood must be at least large enough to contain an experimental physicist and the devices of his laboratory and must allow for enough time for the experiments. With a gradiometer builded by Hughes corporation which has an area of approximately 400 cm2 any one can easily discover if he is leaving in a region of spacetime with a gravitational field or if he is living in an accelerated frame in a region of spacetime free with a zero gravitational field. 9 See [24] for a history of the subject.
12
the mathematical objects appearing in these special relativistic laws are expressed through a set of Lorentz coordinate functions naturally adapted to an arbitrary inertial frame I ∈ sec T M ′ , (M ′ = R4 , η, Dη ) being a Minkowski spacetime (observation 3). Also, if the PLLI would be a true law of nature it could be formulated as follows: Principle of Local Lorentz invariance (PLLI ): Any two LLRF γ and LLRF γ ′ associated with the timelike geodetic lines γ and γ ′ of two observers such that γ ∩ γ ′ = p are physically equivalent at p. Of course, if PLLI is correct, it must follow that from experiments done by observers inside some LLRF γ ′ — say L′ that is moving relative to another LLRF L— there is no means for that observers to determine that L′ is in motion relative to L. Unfortunately the PLLI is not true. To show that it is only necessary to find a model of GRT where the statement of the PLLI is false. Before proving this result we shall need to prove that there are models for GRT were PIRFs are not physically equivalent also.
5
Physical non equivalence of PIRFs V and Z on a Friedmann Universe
Recall that GRT τE is a theory of the gravitational field [4,5] where a typical model τ ∈ M odτE is of the form τ =< M, g, D, T, (m, σ) >,
(30)
where ST =< M, g, D > is a relativistic spacetime and T ∈ secT ∗ M ⊗ T ∗ M is called the energy-momentum tensor. T represents the material and energetic content of spacetime, including contributions from all physical fields (with exception of the gravitational field and particles). For what follows we do not need to know the explicit form of T. The proper axioms of τE are: D(g) = 0;
1 G = Ric − Sg = T, 2
(31)
where G is the Einstein tensor, Ric is the Ricci tensor and S is the Ricci scalar. The equation of motion of a particle (m, σ) that moves only under the influence of gravitation is: Dσ∗ σ∗ = 0. (32) ST is in general not flat, which implies that there do not exist any IRF I, i.e., a reference frame such that DI = 0. Now, the physical universe we live in is reasonably represented by metrics of the Robertson-Walker-Friedmann type [17]. In particular, a very simple spacetime structure ST =< M, g, D > that represents the main properties observed 13
(after the big-bang) is formulated as follows: Let M = R3 × I, I ⊂ R and R : I → (0, ∞), t → R(t) and define g in M (considering M as subset of R4 ) by: X g = dt ⊗ dt − R(t)2 dxi ⊗ dxi , i = 1, 2, 3. (33)
Then g is a Lorentzian metric in M and V = ∂/∂t is a time-like vector field in (M, G). Let < M, g, D > be oriented in time by ∂/∂t and spacetime oriented by dt ∧ dx1 ∧ dx2 ∧ dx3 . Then < M, g, D > is a relativistic spacetime for I = (0, ∞). Now, V = ∂/∂t is a reference frame. Taking into account that the connection coefficients in a (nacs|V) given by the coordinate system in eq.(33) are ˙ ˙ kl , Γk = R δ k Γikl = 0, Γ0kl = RRδ 0l R l i 0 0 Γ00 = Γ0l = Γ00 = 0,
(34)
we can easily verify that V is a PIRF ( according to definition 6) since DV V = 0 and dαV ∧ αV = 0, αV = g(V, ). Also, since αV = dt, V is proper time synchronizable. Proposition 210 . In a spacetime defined by Eq.(28) which is a model of τE there exists a PIRF Z ∈ sec T U which is not physically equivalent to V = ∂/∂t. Proof : Let Z ∈ sec T U be given by Z=
(R2 + u2 )1/2 u ∂/∂t + 2 ∂/∂x1 R R
(35)
where in eq.(35) u 6= 0 is a real constant. Since DZ Z = 0 and dαZ ∧ αZ = 0, αZ = g(Z, ), it follows that Z is a PIRF 11 . All that is necessary in order to prove our proposition is to show that DZ 6= DV. It is enough to prove that the expansion ratios ΘZ 6= ΘZ . Indeed, eq.(4) gives ˙ ΘV = 3R/R, i h ˙ 2 + u2 )1/2 RR˙ + 2R(R ΘZ = , 1/2 R2 (R2 + u2 ) where v = R(
d 1 x ◦ γ) = u(1 + u2 )−1/2 dt t=0
(36)
(37)
10 The suggestion of the validity of a proposition like the one formalized by proposition 3 has been first proposed by Rosen [25]. However, he has not been able to identify the true nature of the V and Z which he thought as representing ‘inertial’ frames. He tried to show the validity of the proposition by analyzing the output of mechanical and optical experiments done inside the frames V and Z. We present in section 7.3 a simplified version of his suggested mechanical experiement. It is important to emphasize here that from the validity of the proposition 3 he suggested that it implies in a breakdown of the PLLI. Of course, the PLLI refers to the physical equivalence of LLRF γs. Also the proof of proposition 3 given above is original. 11 Introducing the (nacs|Z) given by eq.(40) we can show that α = dt′ and it follows that Z is also proper time synchronizable.
14
is the initial metric velocity of Z relative to V, since we choose in what follows the coordinate function t such that R(0) = 1, t = 0 being taken as the present epoch where the experiments are done. Then, ΘV (p0 ) = 3a, and for v and < ξ¯µ > will also have the same outcome in terms of these coordinate charts. Friedman’s statement is not correct, of course, in view of proposition 3 above, for measurement of the expansion ratio of a reference frame is something objective and, of course, it is a physical experiment. However, for experiments different from this one of measuring the expansion ratio we can accept Friedman’s formulation of the PLLI as an approximately true statement. Observation 11. Recall the expansion ratios calculated for V, Z, L, L′ . Now, a
